Quiz

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If you draw a circle with radius 1, and have a ray extending from the origin and intersecting the circle, such that the ray makes an angle \( \theta \) with the \(x\)-axis, we can say that the point at which the circle is intersected by the ray is \((x, y)\). We can define \( \cos \theta \) as the \( x \) value of the coordinate and \( \sin \theta \) as the \(y \) value of the coordinate. Now that we have defined the basic trigonometric functions, we will consider properties of these functions by studying their graphs.

Contents

Sine and Cosine Graphs

In the graph of the sine function, the \(x\)-axis represents values of \(\theta\) and the \(y\)-axis represents values of \(\sin \theta\). For example, \(\sin 0=0,\) implying that the point \((0,0)\) is a point on the sine graph. If we plot the values of the sine function for a large number of angles \(\theta\), we see that the points form a curve called the sine curve:

Similarly, plotting the values of the cosine function for a large number of angles forms a curve called the cosine curve:

We can visualize the relationship between these graphs and the definition of cosine and sine from the unit circle as follows:

Animation courtesy commons.wikimedia.org

How many points of intersection are there between the graphs of \(\sin x\) and \(\cos x\) in the interval \([0, 2\pi]\)?

From the graphs of sine and cosine, it is evident that the number of intersection points in the given range is \(2\). \(_\square\)

Properties

The sine and cosine graphs both have range \( [-1,1]\) and repeat values every \(2\pi\) (called the amplitude and period). However, the graphs differ in other ways, such as intervals of increase and decrease. The following outlines properties of each graph:\[\]

Because \(f(x)\) is strictly positive and \(g(x) \) is strictly negative, there is no intersection point between these two curves. Specifically, there is no intersection point for these curves in the interval \( 0 \leq x \leq 2\pi. \ _\square \)

Relationship between Sine and Cosine graphs

The graph of sine has the same shape as the graph of cosine. Indeed, the graph of sine can be obtained by translating the graph of cosine by \(\frac{(4n+1)\pi}{2}\) units along the positive \(x\)-axis (\(n\) is an integer). Also, the graph of cosine can be obtained by translating the graph of Sine by \(\frac{(4n+1)\pi}{2}\) units along the negative \(x\)-axis. In other words:

Problem Solving

The position of a spring as a function of time is represented by an equation of the form \(p(t) = a \cos bt\). If the spring starts at 3 units above its rest point, bounces to 3 units below its rest point and then back to 3 units above its rest point in a total of 2 seconds, find an equation that represents this motion.

From the context "the spring starts at 3 units above its rest point," we can interpret it as \(p(0) = 3 \), which implies \(3 = a\cos(b\times0) \Rightarrow a = 3 \).

From the context "and then back to 3 units above its rest point in a total of 2 seconds," we can interpret it as the fundamental period of \(p(t) \) is \(2 \). So \(2 \pi \div b = 2 \Rightarrow b = \pi \).